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Math Help - Problem Sequence

  1. #1
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    Problem Sequence

    Be a real number that satisfying: a > 1. Consider the sequence defined by:

    X_1 = 1; X_{n+1} = [\frac{X_n}{2} + \frac{a}{2x_n}]

    1) Prove that X_2 > X_1
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  2. #2
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    As X_{n+1} = [\frac{X_n}{2} + \frac{a}{2X_n}]

    Then X_{2} = [\frac{X_1}{2} + \frac{a}{2X_1}]

    X_1 = 1 so

    X_{2} = [\frac{1}{2} + \frac{a}{2\times 1}]

    X_{2} = [\frac{1+a}{2}]

    If a=1 then X_{2} = \frac{1+1}{2}=1

    as a>1 this implies what?
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  3. #3
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    Quote Originally Posted by pickslides View Post
    As X_{n+1} = [\frac{X_n}{2} + \frac{a}{2X_n}]

    Then X_{2} = [\frac{X_1}{2} + \frac{a}{2X_1}]

    X_1 = 1 so

    X_{2} = [\frac{1}{2} + \frac{a}{2\times 1}]

    X_{2} = [\frac{1+a}{2}]

    If a=1 then X_{2} = \frac{1+1}{2}=1

    as a>1 this implies what?
    Thank you.

    2) Show that if n \ge 2, X_n > X_{n-1} then X_{n+1} > X_n

    It is?
    X_1 = 1 ==>          n = 1
    X_2 = 2 ==> n = 2
    X_n = n
    X_{n-1} = n-1
    X_{n+1} = n + 1
    ?
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  4. #4
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    This is aking for every value after X_2 that the sequence is increasing.

    Can you show that? You could also use mathematical induction.
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  5. #5
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    Quote Originally Posted by pickslides View Post
    This is aking for every value after X_2 that the sequence is increasing.

    Can you show that? You could also use mathematical induction.
    Thanks

    3) Prove that X_1 < a

    I can use my solution in problem 1 ?
    X_2 = \frac{1+a}{2}
    a = 2X_2-1

    X_2>X_1 --> X_1=1 ==> X_2>1
    if X_2=1 ==> a=1 then X_2>1 ==> a>1
    a>X_1
    Correct ?
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  6. #6
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    It is correct ?
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  7. #7
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    Quote Originally Posted by Apprentice123 View Post
    Thanks

    3) Prove that X_1 < a

    I can use my solution in problem 1 ?

    I don't think so. In question 1 X_1 = a and that was given so how can it now be less than 1?
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  8. #8
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    Quote Originally Posted by pickslides View Post
    I don't think so. In question 1 X_1 = a and that was given so how can it now be less than 1?
    You're right.
    How could I prove ?
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